It is YouTube that selects the quality, not Floatbox, so it would be a question for the YouTube community, not Floatbox support.

Any control parameters that YouTube makes available on their iframe embed API can be added to the querystring on the href path of the starting link. For example, Floatbox adds the following default parameters to its YouTube links:

With the caveat that I don't speak for YouTube, am not a subject-matter expert, and last looked into this topic a few years ago, let me offer a couple of observations that should be taken as beliefs, not facts.

In my past testing it was not true that 'auto' always translated to low quality. Quality is (was?) chosen as a function of the video element size and network speed. (There are some indications that the element size needs to be significantly larger than the video resolution for that resolution to be auto-selected.)

YouTube has added, and then subsequently removed, quality controls parameters. I think there used to be 'hd=1', and then later 'vq=...', but neither lasted long.

The YouTube engineers are pretty good at providing the best viewing experience based on the user's device and circumstances. It was a benefit to their users to remove the ability to force high def, especially to users with small screens and/or slow or metered data connections. In my long-ago testing, switching to high def in a context where it wasn't the auto default either failed to present any noticeable improvements to the viewing experience or risked triggering buffer interruptions. There is no one-size-fits-all optimum resolution.

Again, all this is conversation for some YouTube channels. The behaviour is within the youtube embed iframe and is the same wherever that embed is placed on a page. Floatbox just puts some decoration and controls around the outside of the embed.